cavity. For several ports, it describes also the transfer impedances between different ports.
In many practical situations, only one mode is required on each radial port of the parallel plate impedance. This is illustrated in Fig. 3.4c. As will be discussed later in more detail, at relatively low frequencies, the anisotropic modes on radial ports that account for the azimuthal anisotropy, i.e. non-uniformity, of the fields along the ports can be neglected.
The near field model describes the properties of the junction with corresponding electro-magnetic fields and can be relatively complicated for the most general cases. An efficient simulation of these case is therefore difficult. In most practical situations however, the dominant properties can be described with limited effort. The near field can, e.g., be ap-proximated as being azimuthally symmetric. At relatively low frequencies and for cavities filled with a homogeneous, low-loss dielectric, the near field can be described with good ac-curacy by capacitances between the via barrel and each of the parallel plates. Therefore, it is modeled by admittances parallel to the coaxial ports in a topology depicted in Fig. 3.4c.
3.5 Near Field Modeling
In this section, the applicable simplifications are derived and an overview of the most relevant model options is given.
3.5.1 Discussion of Applicable Simplifications
In the following it is discussed how the general description of the junction with two coaxial ports and one radial port constituted by the via and its parallel plate environment can be simplified. The most general model regarding the network parameter block and the considered modal ports is depicted in Fig. 3.5a. On both coaxial ports (one at the top and one at the bottom) several modes are in general required to fully describe the fields present close to and at the port surfaces. By definition, all modes of a port are orthogonal meaning that each corresponds to a solution with a distinct eigenvalue of the corresponding eigenvalue problem. In terms of the fields which relate to the eigen-vectors, no modal field can be represented by linear combinations of other modal fields.
As a first step, it can be made use of the property that not every mode can couple with any other mode if the structure itself is azimuthally symmetric. In the symmetric case only modes with the same azimuthal variation can couple. Therefore, the junction model is split into models for each type of orthogonal azimuthal variation and the coaxial ports for each
TEM-mode coaxial ports anisotropicisotropic mode1mode TM10-mode
(more anisotropic
coaxial ports anisotropicisotropic mode1mode TM10-mode
(more anisotropic modes)
(d)
Figure 3.5:Derivation of a physics-based near field model for vias (a) Most general descrip-tion of the coaxial line–radial line juncdescrip-tion with ordering by physical ports and marking of modes as eitherpropagating (with blue color)ornon-propagating (in red color)for the typical configurations considered throughout this work. (b) As before but with ordering according to the azimuthal anisotropy. (c) As before but with port for non-propagation modes terminated with appropriate impedances. (d) As before but with neglected anisotropic modes on coaxial ports and approximation of the mode termination for the anisotropic modes on radial ports for frequencies well below the cutoff of the higher order modes on the coaxial ports.
3.5 Near Field Modeling variation are grouped. The types of azimuthal variation are isotropy, i.e. uniformity with no angular variation, or sinusoidal variations with different periodicities which correspond to anisotropic modes of different order. It can further be taken into account as a relevant simplification that modes on the coaxial ports with field nodes along the radial direction (such as theTM01-mode [74, Fig. 2.8]) can be neglected because these are well below cutoff.
Both these simplifications are summarized in Fig. 3.5b.
Next, the remaining modes which are below cutoff, but not as far below cutoff as the pre-viously mentioned ones, should be suitably terminated. Each termination must represent the type of field in which the reactive energy is stored in this part of the localized field. The relevant cutoff modes on the coaxial ports are TE-modes which, below cutoff, store the reactive energy in the magnetic field [75, Sec. V.B]. The impedance seen by the junction connected to the coaxial line section is therefore of inductive type and an inductance is, in general, used to model the appropriate termination. For the radial port, the relevant modes are TM-modes. Below cutoff there is an electric reactive field storage that can be modeled by a capacitance. These discussed terminations are used in Fig. 3.5c.
Finally, the following further simplifications are used: For the isotropic mode, all capac-itances of the localized fields are combined to one. For the anisotropic modes, no model for the junction is available in the theory used for this thesis. It is anticipated that the model should be similar to the one for the isotropic modes. Because the inductive termi-nation at the coaxial port can be approximated as short-circuits well below the cutoff of the corresponding modes, the near field model is approximated by a short-circuit of the corresponding radial port. It can be seen that the capacitances on the radial ports are thereby also short-circuited in this approximation.
3.5.2 Near Field Modeling Approaches
In the following, several near field modeling approaches for waveguides filled with a ho-mogeneous medium are presented and an outlook on possible adaptations for modeling of more general cases is given. The near field domain is illustrated again in Fig. 3.6a.
(a)
R:1 Coaxial port 1
Coaxial port 2 Radial Ports
jBa
Figure 3.6: (a) Illustration of the physical near field domain. (b) Equivalent element repre-sentation of a model for the approximate analytical description of the coaxial to radial line junction by Williamson [76] inside the area marked with the dotted line. Drawing for the case with two coaxial ports adapted from [76]. (c) Equivalent circuit applied in the second, mostly analytical model as described in last paragraph of Section 5.3.2. The near field is modeled by Yeff parallel to each coaxial port. (d) Corresponding representation of the FDFD model.
Figures and text adapted from [13].
3.5 Near Field Modeling
Relevant Modeling Approaches for Homogeneous Fillings
The first type of near field model uses a quasi-static modeling. For comparatively low frequencies, the field parts in the junction that are in general identified as cut-off modes are in this case identified as the (static) capacitances between the via barrel and the reference planes. The static definition results in a frequency independent value for all cases where it represents a sufficient approximation. For this approximation, the static capacitance definitions require the barrel and the reference planes to have each a uniform potential.
This type of approximation has been studied for PCB vias with various techniques [77–83].
The second type of near field model takes into account the wave properties. The models neglect fields with azimuthal variations but include the effects of reactive field of the cutoff modes and an impedance transformation at the junction. The model of this type that is considered in the following is the model first proposed in this form by Williamson [76,84,85].
An equivalent circuit is depicted in Fig. 3.6b. The admittances jBa, jBb, jBc, and jBd model parts of the capacitive near field, while the transformer models the impedance transformation between radial port and coaxial ports. For many practical applications in PCBs, it has been found that several simplifications are possible, e.g., the transformer ratio is relatively close to 1. A related near field model is known as the intrinsic via model and has been discussed in [63,86]
3.5.3 Outlook on Adaptations for Layered Fillings
For the case with layered fillings, the detailed discussion of all adaptations is given in Chapter 5. A short outlook is given in the following.
The first type of model assumes that an admittance parallel to each port provides a near field model for the quasi-static case. In the model depicted in Fig. 3.6c, the admittancesYeff represent effective values of via barrel to plate conductance and capacitance. Due to the frequency-dependent complex permittivity of the silicon ε =ε0εr−jσsi/ω, the values are not constant with frequency as is the case for a medium without conductivity. A relatively simple approximation based on static field computations is presented in Section 5.3.2. More accurate results could be obtained by adapting a contour integral method for quasi-static capacitance calculation as in [16, 87]. This has not been done but instead development of the full-wave approach has been pursued.
The full-wave approach used in this work performs a local full-wave modeling of the area around a via barrel using a finite-difference frequency domain (FDFD) method in cylinder coordinates. This is the main subject of Chapter 5. As labeled in Fig. 3.6d, this local
full-wave model gives results for both the localized near field and the propagating far field in a modeled region around the via barrel. In order to enable the comparison of all near field modeling approaches, one option is to add radial line segments to the modeling approaches which model only the localized field as is depicted in Figs. 3.6b and 3.6c.